Quantum Ising chain with time-averaged work in linear response theory

For systems performing a weakly isothermal process, the decorrelation time dictates how fast the relaxation function decorrelates. However, like many other thermally isolated systems, the transverse-field quantum Ising chain presents an ill-defined decorrelation time. On the other hand, the Kibble-Zurek mechanism uses a heuristic relaxation time to achieve its famous scaling. The problem however of having a well-defined decorrelation time, derived from first principles, agreeing with the Kibble-Zurek mechanism is still open. Such a solution is proposed here by measuring the work using the time-averaged relaxation function of the system, which offers a new and well-defined decorrelation time for thermally isolated systems. I recover with this the Kibble-Zurek mechanism in the finite-time and weak driving regime, and new features in the slowly-varying one. The gain in control over the system in such a distinction is desirable for potential applications.